I have a second order system with the following step response: $$G_2(t) = a_o + a_1 e^{-t/\tau_1} + a_2 e^{-t/\tau_2}.$$
I would like to know what the settling time is through dominant pole approximation. Assuming the settling time, $t_s$, is reached at 95%, for a similar first order system it is common to say $t_s = 3\tau$. In the dominant pole approximation you can use the smallest pole (i.e. the largest time constant, $\tau$) to estimate the settling time, using the same formula as a first order system.
However, the conditions in which this approximation is valid were difficult to find without attempting my own mathematical proof.
I have seen one source say that the dominant pole approximation is only valid when: "Dominant pole time constant is 5x larger than other poles" for a generalised system (with no proof):
$\frac{A}{(s+p_0)(s+p_1)...(s+p_n)}$
For my system I assume there could be another condition ' $a_1/a_2 < threshold$ ' as well?
Or do I need to define an allowable region for a function with x and y axes, $a_1/a_2$ and $\tau_1/\tau_2$ since there are two conditions?
Thoughts?